Advanced Topics in Term Rewriting

Abstract

Library of Congress Cataloging-in-Publication Data Ohlebusch, Enno.Advanced topics in term rewriting / Enno Ohlebusch. p. cm. Includes bibliographical references and index.1. Rewriting systems (Computer science) I. Title.

Models for Logics and Conditional Constraints in Automated Proofs of Termination

“…The material in Section 2 can be thought of as a generalization and extension of the intepretation method for proving termination of Term Rewriting Systems (see, e.g., [17,Section 5]). The interpretation method uses ordered algebras which are algebras A with domain D including one or more ordering relations D , D , etc., satisfying a number of properties (stability, monotonicity, etc.).…”

“…The interpretation method uses ordered algebras which are algebras A with domain D including one or more ordering relations D , D , etc., satisfying a number of properties (stability, monotonicity, etc.). Such relations are used to induce relations , on terms which are then used to compare the left-and right-hand sides and r of rewrite rules → r. The targeted rules in such comparisons and the conclusions we may reach depend on the considered approach for proving termination (see [17,Sections 5.2 and 5.4], for instance). In our setting, orderings are introduced as interpretations of computational relations (e.g., → and → * ), and we do not require anything special about them beyond their ability to provide a model of the theory at hand.…”

“…An interpreter for a logic L (for instance the logic for Conditional Term Rewriting Systems (CTRSs) with inference system in Figure 1) is an inference machine that, given a theory S (e.g., the CTRS R in Example 1) and a goal formula ϕ (e.g., a one-step rewriting s → t for terms s and t) tries to incrementally build a proof tree for ϕ by using (instances of) the inference rules B1,...,Bn A ∈ I(L) of the inference system I(L) of L. Then, S is operationally terminating if for any ϕ the interpreter either finds a proof, or fails in all possible attempts (always in finite time). In this setting, practical methods for proving operational termination involve two main issues (see [13] and also [17] for CTRSs): (1) the simulation of the (one-step) rewrite relations → and → * associated to a CTRS R and defined by means of the inference system in Figure 1; and (2) the use of (automatically generated) wellfounded relations to abstract rewrite computations and guarantee the absence (Refl) t → * t (Cong)…”

From Conditional to Unconditional Rewriting

“…We recall some basic notions of conditional rewriting, referring the interested reader to [14] for more details. An (unsorted) signature Σ is a finite set of operational symbols, each having zero or more arguments.…”

“…A large body of literature has been dedicated to transformations preserving only certain properties of CTRSs, e.g., termination and/or confluence. We do not discuss these here; the interested reader is referred, e.g., to Ohlebusch [14].…”